Can not find spiral

[MrAl]

Hello,

I am looking for a special kind of spiral, not the ordinary type
and maybe not a spiral at all as it is probably composed of
two distinct parts...a spiral and a solution to another equation.

This 'spiral' (i'll call it that for lack of a better term) is formed when
a long object such as a piece of tape is wrapped around a circular
post, with layers going one on top of the other. I say tape rather
than cord because the layers will lay flat on top of one another as
the winding progresses possibly with thousands of turns.

Now it might be tempting to call this an Archimedean Spiral, but
it is not because the tape as viewed from the side does not gradually
build on the bobbin (post) as the tape thickness, but starts out
as an almost complete circle, then sort of 'hops' up to the next
layer when it gets to the rather abrupt start of the winding, then
gets progressively smoother and smoother.

Let me try to explain this winding, from a side view of the layers.
The first layer starts with one end of the tape pressing against
the post (or bobbin), and that has an edge equal to the thickness
of the tape. The tape is then wound around the circular post until
it almost comes to the start of the tape (the first end) and then
it slopes up a tiny bit, but as the tape is wound over that end,
it has to bend over the corner of the tape end, and so as the next
layer is started the place where the tape meets the end of the
tape on the first layer and the elasticity of the tape causes it
to bulge up just a little, and that pulls the part of the tape just
before the tape end to bulge up a little too, so what we get
intead of a slope and then another circle is a sort of ocean
wave looking curve, until some point after that the curve
smooths out again to a circle. The third layer would have
the same characteristic, but would be a little smoother overall,
and the fourth smoother, etc., until some very many turns
later it starts to look like an almost perfect circle.

The equation would have to include the elasticity of the tape
or some way to approximate this to account for the wave
like shape at the layers over the tape end (actually the tape start).

I have approximated this action to a pretty decent accuracy, but
i would like to find out if anything has been done before like
this, and if so, compare my results to that.

Thanks.

[jwax]

Hi MrAl- I'm curious as to what exactly you're looking for.
Are you after the exact precise equation for this distorted spiral?
How is the "elasticity" of the tape expressed? Are their standard units for that?
Also, is the point to determine the length or diameter of the spiral?
I use this for calculating paper on a roll, ignoring the overlap "bump" of course-
http://www.cutsmart.com/pages/roll-length.html
John

[Bob Scott]

HaHa!

I've often wondered as I entered the room, is there enough left or should I get another roll first?

Or has this got something to do with making capstans or drive idlers larger? I seem to remember that years ago I doubled the circumference of the capstan in my reel-to-reel tapedeck in an attempt to increase tape speed to 15 ips by wrapping tape on the capstan. It was the bump caused by the thickness of the other end of the roll that was a problem.

Bob

[Dean Huster]

If you've been trying a search for the math, try using "helix" rather than "spiral". Helix is the analytical geometry term for what we usually call a spiral.

Dean

[MrAl]

Hi again,


First off, i really appreciate the responses. I thought maybe i
was going to get stuck with no incoming ideas at all. Even some
ideas that dont seem appropriate sometimes makes you think
about something you didnt think about before, and then end
up with a better result.


jwax:
Well the elasticity might be expressed in terms of Young's Modulus.
This is often used to determine how something bends using formulas
for however that object bends in space.
Obviously the tape stretching is going to have something to
do with it too.
That link was interesting, but they are in fact using the mean
diameter formula to calculate the result. That is, they take
the mean diameter, multiply by pi, multiply by the number of
turns N=(outer dia-inner dia)/thickness/2, and divide by 12 to
get the answer in feet.
The difference between the A. Spiral and the Mean Diameter
Formula is very small when the tape thickness is very small,
so that is probably a very good approximation. What i am
looking for is a bit different however, perhaps more theoretical
than practical, just as a means to understand the error a little
better. I suppose the idea could be taken further though.

Bob:
Yes, that bump effect is what i want to calculate, or at least
the effect it has on the total length of tape, even if it is
very small (which im sure it is).

Dean:
I tried helix but they keep stating that helix is a space curve.
What i am looking at here is a 2 dimensional curve, if we can
actually call it a curve that is, as it is irregular.
As the tape piles up, it stays lined up with all the tape under
it just like a roll of electrical tape.



I am starting to think that after some smaller number of turns
(possibly 10 or so) that the roll smooths out enough to very
closely approximate an Archimedean Spiral. That would mean
most of the error occurs near the start of the roll. It would
be nice to be able to calculate this whole 'spiral' even with
the irregularities though.

[CeaSaR]

Al,

From your description, it sounds like you want to introduce a minute
reverse curve (main winding into the "bump") into a second reverse
curve (out of the "bump" back into the main winding) that then becomes
concentric with the main winding again. The first set of reverse curves
would be very sharp with each successive set becoming more and more
gradual until after approximately 10 - 20 sets the reverse curves are so
negligible as to be ignored. Perhaps the quickest way to calculate this
error would be to do the first set (sharpest transition), and the last set
(smoothest transition before maintaining spiral/helix), and then pro-rating
the remaining number of calcs between the min and max.

Therefore the first set would have central angles very close to 90 degrees
while the last set would have central angles very close to 180 degrees.

With mylar tape (I am assuming from your description above this is
probably related to video / audio tape) you might pick up an extra mm
or 2, hardly enough to affect any length calcs in relation to time calcs
(your previous threads). Obviously, with any thicker tapes you would pick
up more length, but those types of tape (sticky backed) the advertised
length is only an average anyway.

CeaSaR

[ecerfoglio]

If you've been trying a search for the math, try using "helix" rather than "spiral". Helix is the analytical geometry term for what we usually call a spiral.
Dean

Dean:
I tried helix but they keep stating that helix is a space curve.
What i am looking at here is a 2 dimensional curve, if we can
actually call it a curve that is, as it is irregular.
As the tape piles up, it stays lined up with all the tape under
it just like a roll of electrical tape.

Mr Al is right, a helix is a 3 dimensional curve (a screw's thread is an helix), and a spiral is a 2 dimensional (contained in a plane) one.

The problem is that, ussually, we use "spirals" to call some helix shaped objects - like a car's suspension springs.

[Bob Scott]

Bob:
Yes, that bump effect is what i want to calculate, or at least
the effect it has on the total length of tape, even if it is
very small (which im sure it is).

I would think that the initial "bump" would be parabolicly shaped with the beginning of the tape located at the apex, but the exact starting shape isn't important because successive winds will pull the bump down flat due to take-up tension in the tape deck. The end of the first layer will have to undergo a flat linear rise up from the level of the hub to the start point of the second layer.

Assuming frictionless tape, successive winds will apply more than enough pressure to flatten the bump until it is just a kink, just like the force multiplication due to successive winds in a block and tackle. The tape linear tension stays the same for the entire length of the tape, but the pressure towards the centre of the hub is multiplied. From there on you just need geometry to predict what the shape of the windings looks like, with the layers over the kink becoming more gently curved. Use the kink point as the centre of that radius. Use the hub centre as the second radius, and a ruler to draw the linear ramps between the circular curves. I don't think that this short flat ramp section goes away in successive layers no matter how many layers there are, but stays a constant length for all layers.

However, magnetic tape does have friction. I wouldn't think it would be documented. Check out "block and tackle" in Wiki.

[MrAl]

Hi Bob,

I'll have to think about this a bit.

[CeaSaR]

If you really want to stop and think about it, what you are actually looking
at is a series of concentric circles with a slight "bump" where the first
circle (tape) begins. A spiral indicates a central starting point with each
point of the compass being further away from the center than the last
point, which is not really the case here. That is, if the hub is machined /
manufactured as a circle and not a spiral base. If the hub were machined
as a spiral base, then the bump should be negated because the tape
should be wound correctly along the spiral with no bump.

Just more food for thought.

CeaSaR

[MrAl]

Hi Ceasar,

Yes, if the hub was specially made it could have the end sunken
down below the next winding. For this thread i am assuming a
perfectly circular hub however. This means the start of the tape
presents a sort of 'step' to the next layer as i was describing.

Im going to draw a picture of this next i think so that the
idea of what is going on becomes clearer.

[CeaSaR]

I understand completely and am assuming a perfectly circular hub also,
as a spiral based hub would be overkill for a manufacturer, likely more
expensive to mass produce accurately. I just wanted to throw my
observations into the fray for consideration.

CeaSaR

[MrAl]

Hi Ceasar,

Yes, and thanks for doing that. I like to hear different ideas like this.

[dyarker]

I would think the "bump" can't disappear ... exactly ... sort of ...

The radius of the bump would increase with each additional layer of tape, due to the thickness of the tape (very thin, but not zero).

Keep winding tape till till the radius of the bump equals the radius of the tape at that layer and I guess you'd get an egg shape that can be measured by micrometer but can't see with naked eyeball.

[MrAl]

I would think the "bump" can't disappear ... exactly ... sort of ...

The radius of the bump would increase with each additional layer of tape, due to the thickness of the tape (very thin, but not zero).

Keep winding tape till till the radius of the bump equals the radius of the tape at that layer and I guess you'd get an egg shape that can be measured by micrometer but can't see with naked eyeball.

Hi again,

Well, i think you are right about that. Even though the bump is
small it probably propagates through the layers, unless of course
there are so many layers that the subsequent layers absorb the
extra height by way of their compressibility.
The other thing i was thinking was that the next layer presses 'down'
on the bump, which pushes it down and so the tape that makes
up the bump gets pushed a little tighter on the reel, which makes
the bump slightly smaller.
These and other effects and questions come up so that's why i
was hoping there was something out there already written on this.
It's hard to start from scratch.

Here's a starter list for what would have to appear in the
calculation for the layers:

1. Young's Modulus for that material.
2. Tension on the tape as it's being wound.
3. Of course all the physical dimensions like spool diameter.
4. Temperature, although i think i would be happy with 25 deg C alone.